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03a - < ∞ and φ 6 M then Z φ ◦ | f |)d μ 6 φ t μ...

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Modern Analysis 2 Test 3 Answer TWO questions. 1. Let ( f n ) n =0 be a sequence of measurable functions. Prove that each of the following sets is measurable: (e) E = { x : f n ( x ) > e n } ; (f) F = { x : f n ( x ) < 0 for finitely many n } ; (h) H = { x : n > 0 f n ( x ) = 1 2 ( f n +1 ( x ) + f n - 1 ( x )) } ; (i) I = { x : lim n f n ( x ) Z } . 2. Let f : X R be measurable and φ : [0 , ) [0 , ) be increasing. Prove that if t > 0 then Z ( φ ◦ | f | )d μ > φ ( t ) μ {| f | > t } and that if also μ (
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Unformatted text preview: ) < ∞ and φ 6 M then Z ( φ ◦ | f | )d μ 6 φ ( t ) μ ( X ) + Mμ {| f | > t } . 3. Say f n μ → f iﬀ ( ∀ ε > 0) μ {| f-f n | > ε } → . Let μ ( X ) < ∞ . Deﬁne d ( f,g ) = Z | f-g | 1 + | f-g | d μ. Show that f n μ → f ⇔ d ( f,f n ) → . 1...
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